As the company holds the stakes, if it wins, it simply retains and uses the $9.29 or $15.06 which the individual lost by not dying. If it loses it pays the heirs of the insured its bet of $990.71 in one case, or $984.94 in the other, and also returns the stake of $9.29 or $15.06 which it did not win, making up the even $1,000 of the policy in either case. No matter what the number of such bets, if the expectation of the table is exactly realized, the bets of the insured exactly pay the losses of the company, and it has nothing left at the end of the year. Such bets of course, could not be settled till the end of the year, and to make itself sure against an excess of deaths, which might possibly occur, out of a small number, the company must either insist upon the individual increasing his bet, so as to diminish its own liability to loss, or it must have a capital that can equalize the fluctuations of loss for successive years, and establish a probability of profit or surplus, by avoiding all risks worse than the average, and enticing in those that are better. It must also add enough for the expenses of the office. Practically, the experience of the companies shows, that even with small numbers, there is no necessity of increasing the individual's bet beyond the requirement of the table. Hence, so far as the insurance itself is concerned, he needs to pay or stake, at the beginning of the year, only what at the assumed rate of interest, will amount to the tabular bet at the end of it, plus his proper contribution for expenses to be incurred; and this addition, beyond doubt, should be proportioned either to his bet, or to the face of the policy. The sum to be paid in advance, without the addition or "loading" for expenses, is called the natural or tabular net premium. With interest at four per cent., it is $8.93 per $1,000, at thirty-five; and $14.48 at forty-nine. If the policy is for a single year, any addition to either, beyond what is necessary for the working expenses, increases the bet of the individual, and diminishes that of the company. If at thirty-five, the party being an average risk, instead of paying $8.93 and a given addition for expenses, should pay $100 and the same addition, his bet would be $104 against the company's bet of $896. And should the company win, it would win the whole $104, out of which the clear business profit would be more than the difference between $104 and $9.29, because what it costs (as a matter of average) to carry the risk in the two cases, must be proportioned to the company's exposure to loss in each. Hence the clear profit of winning the latter bet, compared with winning the former, is 104—

life insurance, and if it were, it would not be based on any table of mortality. So much for a policy covering a single year.

When a policy covers a number of years, the premium may be much increased, but the bet is decreased, for a very good reason, which will presently appear.

If a company should agree to make, say fifteen successive tabular yearly bets with one thousand persons of sound health, all aged thirty-five, charging for each year the net natural premium, increasing from $8.93 to $14.48, plus expenses, per $1,000 of policy, it would, or might meet with this difficulty. The healthiest lives, having nothing to lose thereby, might before the end of the fifteen years, fail to deposit their stakes; while those of failing health, more likely to die within the term, would persist. And thus, without any extraordinary mortality, the company might become bankrupt, by the dropping out of the healthier co-insurers. The only remedy for this, hitherto adopted, although on the whole effective, is not uniformly so, and it is far from being always equitable when it is effective. It is to commute the series of natural increasing net premiums 'into an exactly equivalent single premium, or series of equal annual premiums.

The result of this, in regard to the sort of policies most pressed upon the public, is to put into the hands of the company, at once, a good deal of money, not immediately, or for a long time, required for the payment of losses; and, by the terms of the policy, any policyholder failing to pay his premium when due, forfeits to the company whatever he may previously have contributed to this overplus or reserve for future use. This is a good and sufficient—and often more than sufficient-security for persistence, for some policies, but not for others, as will better appear by and by.

The method of commuting is, to discount at compound interest each future natural net premium, and multiply each of these results by the fraction expressing the tabular chance of the party living to pay it. The sum of these discounted values, with the premium due in advance for the first year, is the net single premium. This divided by the present value of one dollar per annum, the first payable in advance, for fifteen yearly payments, discounted both by interest and mortality, as above, is the equivalent level net annual premium. For example: the net natural premiums to insure $1,000 per annum, from thirty-five to forty-nine, inclusive, commute at four per cent. into a net single premium of $115.33; and this, divided by ($10.85) the present value of an annual dollar, subject to the mortality discount, gives the level net annual premium, $10.63. The only

difference between this fifteen year policy and an endowment insurance policy, payable to the party himself at fifty, or on previous death, is, that the net natural premium of the last year of the latter must be the present value of $1,000, payable certainly at the end of the year, because the claim must then be paid, be the party dead or alive. Hence, instead of $14.48, it must be, at four per cent., $961.54. The difference,-$947.06,-payable certain at the end of fourteen years, discounts, at four per cent., to $546.90, and this

is further reduced by the mortality discount,

70580
82581'

to $467.43. If

we add this to the net single premium for the term insurance$115.33,—we have $582.76 as the net single premium of the endowment insurance policy. And this divided by the same present value of the annual dollar, gives $53.72 as the net annual premium of the same.

It is from not understanding the effect of this increase of the commuted over the natural premium, to $10.63 in one case, and to $53.72 in the other, that most of the mystification of the policyholders comes, and perhaps more than half of the bad management of the companies. Its effect is, to throw part of the insurance on to the insured party himself. By the law, both of necessity and the State, the sum of the two bets is no longer equal to the face of the policy. The individual does not bet his whole net premium on the insurance of the year, as in the case of the single year policy, and, if he loses, the company cannot use the whole of it to pay its losses with. It can use only what he bets and what it wins. The problem is to find what he does bet. The solution is this, and it is perfectly general, embracing all commuted premiums, single or annual, whether of term or endowment insurance policies, including the whole-life policy, which is, in effect, endowment insurance. The company's bet is simply what it is exposed to lose by the death. If, by paying the natural premium of $8.93 in advance, amounting to $9.29 at the end of the year, the individual bets that sum, the company bets, and is exposed to lose, $990.71. But if, by a commutation of the natural premiums for the whole contract, the individual pays $53.72 in advance, amounting to $55.87 at the end of the year, the company is exposed to lose, by the death happening in that year, only $944.13. This is its bet for that year. And, by the conditions of the commutation, or necessity of the case, the individual's bet is reduced in the 944.13 same proportion. It is 990.71

X 9.29

= 8.85.

Hence, if the

company wins, by the individual being alive to pay his second

premium, it wins, and can use out of the $55.87, only $8.85, leaving $47.02 as a reserve for future use, a necessity of the commutation. But, if the individual wins by dying, his claim, under the policy, is in reality as follows:

made

up

The most important thing for a policy-holder to know about his policy is, how much he is insuring himself under it. Because, what he insures himself, the company does not insure. It is merely holding that sum for him in trust, just as if it were a savings bank. He knows what it would cost him to have it in a savings bank,—on the average 100 of one per cent. per annum. Then, if he knows what his whole policy cost him, by deducting the proper cost of the self-insurance he will find out what is the actual cost to him of the insurance which the company is doing for him, in any year, as compared with what it ought to be by the tabular assumptions.

26

As the insurance done by the company, in any year, under a commuted net premium, is always proportioned to its bet on that year, or exposure to loss, compared with the bet, or loss it would be exposed to, if only the natural net premium due to the present age were at stake on the part of the insured, it is quite easy, with the mortality table before one, to calculate, at any interest, the consecutive insurances by the company on any policy from beginning to end. And the self-insurances are always simply complementary to the

insurances.

In view of the following extract from the mortality table recognized as the standard in this State, the calculation of the insurances and self-insurances of two policies, at two rates of interest, which follows it, will be readily understood, and will serve as a text for some brief remarks, which the Committee consider of great practical importance. As the decimals are not carried beyond the cents, the results are not exact, but sufficiently so for illustration.